   Chapter 1.2, Problem 25E

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# Let f : A → B , where A and B are non empty, and let T 1 and T 2 be subsets of B .Prove that f − 1 ( T 1 ∪ T 2 ) = f − 1 ( T 1 ) ∪ f − 1 ( T 2 ) .Prove that f − 1 ( T 1 ∩ T 2 ) = f − 1 ( T 1 ) ∩ f − 1 ( T 2 ) .Prove that f − 1 ( T 1 ) − f − 1 ( T 2 ) = f − 1 ( T 1 − T 2 ) .Prove that if T 1 ⊆ T 2 ,   t h e n   f − 1 ( T 1 ) ⊆ f − 1 ( T 2 ) .

(a)

To determine

To prove: f1(T1T2)=f1(T1)f1(T2) for f:AB, where A and B are nonempty sets, and let T1 and T2 be subsets of B.

Explanation

Proof:

Suppose af1(T1T2). Then there exists bT1T2 such that f(a)=b.

Hence, bT1andf(a)=b. If bT1, then af1(T1)f1(T1)f1(T2) and if bT2.

Then, af1(T2)f1(T1)f1(T2). Thus f1(T1T2)f1(T1)f1(T2).

Now, suppose that af1(T1)f1(T2)

(b)

To determine

To prove: f1(T1T2)=f1(T1)f1(T2) for f:AB, where A and B are nonempty sets, and let T1 and T2 be subsets of B.

(c)

To determine

To prove: f1(T1)f1(T2)=f1(T1T2) for f:AB, where A and B are nonempty sets, and let T1 and T2 be subsets of B.

(d)

To determine

To prove: T1T2,thenf1(T1)f1(T2) for f:AB, where A and B are nonempty sets, and let T1 and T2 be subsets of B.

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